Volcano Day
by wholockian729
Summary: This story was written as a trig project. A Post-Reichenbach epic where John and Sherlock get wisked away to the Reichenbach isles where they must work through a series of labyrinths and problems. They have 24 hours to overcome Sherlock's knowledge deficiency in Trigonometry and related concepts before a volcano erupts, taking Mrs' Hudson's and our favorite duo's lives.
1. Chapter 1

The Empty Hearse

Try as he might, John could type no more. He had been staring at the screen for ten minutes, thinking about whether or not he wanted to blog this case. It would be his first posting in so long. His first story since his partner in crime solving had come back…from the dead.

It had been just under a month since Sherlock had returned. Just over one year since he had jumped from the roof of St. Bart's, plunging like an unfledged bird to the sure death waiting below. After all, falling was just like flying, but with a more permanent destination. Sherlock had been beaten by the insane James Moriarty, and it had caused the most destitute of repercussions for the still-healing army doctor. The only good that had come of the situation was the death of the twisted, evil, spider-like man. It should have killed Sherlock as well, but somehow that genius man had managed to survive. John still didn't understand how.

Things were just starting to get back to normal after the year spent apart. Luckily, John had been too depressed to remove any of Sherlock's things. Instead, he had left the flat almost exactly as it had been before the whole Reichenbach mess had started, save for the body parts in the fridge, and the more caustic looking experiments. John had just gotten Sherlock back, and wasn't quite sure he wanted to share his newly resurrected mate with the world. Though Moriarty himself was dead, there were still people loyal to him out there. Undoubtedly, some were scanning John's blog for word of the detective, just in case the snipers had been duped into believing that Sherlock was dead. John didn't want to instigate any other danger from what remained of Moriarty's web.

John groaned and shut the laptop. He sank back into the couch and covered his face in his hands. Life with Sherlock had become so complicated, yet it remained so easy. The detective's behavior was even more erratic than before. He had night terrors that John could easily hear from his upstairs room, but refused to tell John about them or even admit that he was having them. When John could actually get Sherlock to sleep, it was often interrupted by these dreams. Neither of them got any sleep when this happened, so both were often irritable.

It bothered John that Sherlock wouldn't come to him for help. John had suffered under similar conditions when he got back from Afghanistan, before he had moved into 221B. John knew it shouldn't bother him, as Sherlock was often reclusive, and never like showing any emotion. John couldn't, however, suppress his distaste at the fact that Sherlock didn't trust him enough, as his best friend, to let John help him. Sherlock had always been frustrating, so John could deal with it. In most other ways, life was no different than the Pre-Reichenbach simplicity. Sherlock would experiment, John went to the surgery, Sherlock drank all the milk, and John replaced it. Life was finally getting back to normal.

John gave a small chuckle, as he realized that nobody, save him, would likely call any interaction with Sherlock Holmes "normal", yet here he was, describing life with him as normal. It was insane, but John knew that since he had been exposed to this life, he could never live any other way. As if to prove this thought, his mind drug up memories of after The Fall, when everyone had been urging him to move on, to get over Sherlock Holmes. John hadn't been able to let go of his best friend. He then thought to that night Sherlock had finally returned.


	2. Reunion

John walked into the flat, returning from his early shift at the surgery. He went immediately to the kitchen to put away the bags from Tesco's. As he walked into the kitchen, his whole demeanor shifted from the happy facade he had taught himself to wear when in public, to the repressed, tired self he had been for the past year. He flinched at the sudden realization. It had been a year since Sherlock had taken his own life. John still didn't know why. Surely Mycroft could have restored his reputation? Surely Sherlock, who gave no thought to what others thought of him, wouldn't have been so disgraced by Moriarty's exploits that he had truly felt the need to end his life?

John shook his head to chase away these unwelcome thoughts from his mind. He had only returned to the public in the past two months. He refused to go out any earlier, as until that point, Sherlock had been plastered everywhere, the media still having a field day with the apparent deceit of the consulting detective. Awful things had been said about his late best mate. His reputation as a genius man was reduced to a deceitful snake, willing to do anything, hurt anybody to put himself on top. The media had finally gotten bored of Sherlock after a few months, and the news about him lessened considerably. That was until new information came forth, undoubtedly from Mycroft, proving that it had all been a lie. Proof about the truth behind Moriarty came out, and Sherlock's name was cleared. John was finding, however, that many preferred to shun the reality, and accept the lie Moriarty had spread. It just went to show how brilliant Moriarty's plan had been. He took one lie, wrapped it up in the truth of Sherlock's life, and then fed it to the public. It was even easier to digest because no one wanted to feel inferior, which Sherlock invariably made them feel. Sherlock had fought hard, but even in death, Moriarty won.

John shook his head to chase away these unwelcome thoughts from his mind. He hated thinking of Moriarty for too long, as it often brought back the nightmares that had haunted him the nights after the fall. Horribly vivid images of Sherlock falling from great heights, then splattering against the pavement, joyous laughter ringing from above; it filled his head each and every time he closed his eyes. Each time Sherlock landed, it was a bit different. John hadn't seen the impact, he only witnessed the aftermath. People rushing to see the spectacle, paramedics pouring from the building… his best friend, bloody and battered; this always left John shaking uncontrollably. Any reminder of that once great man could set John off.

He had only returned to the public in the past two months. He refused to go out any earlier, as until that point, Sherlock had been plastered everywhere, the media still having a field day with the apparent deceit of the consulting detective. John was just now getting his old life back. This new life however, was missing the one crucial aspect that made his life bearable after Afghanistan. The consulting detective who had made his life interesting was now gone, and life was dull once more. John chuckled at this thought. Sherlock had rubbed off on him. Everything was dull.

He wasn't sure what made him turn around. He still had groceries to put up and tea to make. But at that moment, he felt the need to turn on heel and gaze around the rest of the kitchen. Later he would claim that he knew the other occupant in the kitchen was there the whole time, but both men knew that was a lie. Perhaps he had subconsciously noted the accidental, yet still quiet clink of test tubes. Or maybe he felt the warmth return to the flat, which had been cold and desolate since… Sherlock had died.

Regardless of the reason, John obeyed his instincts, turned to stare at the kitchen table, and promptly fainted, falling flat on his face, after a gasp of surprise escaped his mouth. For there, sitting calmly at the kitchen table, was a tall, lanky man with curly brown hair. There, at the table, was Sherlock Holmes, piercing stormy-blue (at the moment) eyes analyzing the experiment laid out in front of him. Just like before. As if nothing had changed. As if he hadn't died over a year ago.

Sherlock glanced up at the sound of John's gasp, eager to explain everything to his best mate. He frowned when John crumpled to the ground, put off a bit by this inconvenience. He wanted to tell John now, but apparently John wasn't ready to listen. Sherlock huffed, pulled his usual scowl, and then turned back to his experiment. He didn't have much left to do, so it was finished quickly. Now he was left with nothing to do until John decided to wake up and let Sherlock explain. He glanced around the kitchen, looking for a distraction. His eyes fell on John's unconscious body, and a pang of regret ran through him. Sherlock had done this. John would probably have a headache when he woke, as he had hit his head fairly hard when he fell.

Unfortunately, while Sherlock had an extensive knowledge of human anatomy and physiology, he was mostly clueless when it came to administering medical attention. He thought back to what John did when Sherlock was hurt, and decided on a course of action. He quickly fetched a cushion from the couch, and placed it under John's head, hoping to lessen any pain he might have later from the hard floor. He then went to the closet where he knew John would keep his medical supplies and pulled out a Band-Aid. He knew people generally used these when hurt, so he unwrapped it, observed it a bit, trying to deduce the best spot for it, and settled for placing it on John's forehead, where the bruise would most likely appear in the morning.

Sherlock sat back on his heels to wait for John to wake. As he waited, he battled the ever growing boredom by observing and deducing the causes of the physical changes that John had undergone in the year they had been apart. His hair had recently been cut. This made Sherlock frown, as he knew John kept a strict haircut schedule, and his last haircut should have been four months ago, with his next cut coming up in two months. Why had he waited? Had he become reclusive, and only recently started venturing out into public? It was the only explanation that fit all the facts. John was a creature of habit, and only something extreme would throw him off. What had been extreme enough to keep John from keeping his hair tidy, a habit instilled in him from his army days?

_Well, what has changed recently?_ Sherlock asked himself. _It is obvious from his shoes and his hair that he hasn't been out much. The pattern on the soles of his shoes are hardly less defined than when I last saw him, and there is a lack of dirt older than, I'd say, two months on his shoes. So he has only rejoined the public within the last two months. What changed that would have brought him out of hiding? He wasn't mourning me that whole time, was he? My "note" was supposed to prevent that. Are thoughts of me still painful? Seems more likely. I know Mycroft cleared my name and got it out of the papers recently. It would be hard for John to walk about London with everything talking about me, if he was still caught up on me. Reminders of me could hurt him enough to keep him in the flat, could they? Even if they did, why would he stay in the flat, the biggest reminder of all? It must be sentiment. _

Sherlock left that train of thought and returned to John. The lines on his face had deepened, aging him past the year they had been apart. Had some hardship befallen him? He frowned to himself as John finally roused. Groggily, John asked "Sherlock?"

"Yes, John. Obviously, it is me. I am here." The younger man said in his usual condescending tone.

"Wha- How-"John stammered as he scrambled as far away from the other man as the physical constraints of the small kitchen would allow. He huddled against the cabinets, staring in wide eyed surprise at the supposedly dead consulting detective.

"John, I can explain." Sherlock said cautiously, not wanting to startle his friend again.

"No. No! You're dead!" John whispered furiously, almost to himself. He cautiously lifted his gaze to the other man's eyes, and their gazes locked. John was stationary just a moment longer, before he started chuckling.

"I get it, Sherlock." He said the laughter building with each word. Sherlock's expression filled with surprise and joy.

"You do?" He asked. He had always known John was smart, but had never pegged him as perceptive enough to understand this.

"Of course!" John said, his expression dropping "You're one of those hallucinations the therapist warned me about." His whole body sagged into a lump on the floor. Sherlock was startled. He had anticipated anger, joy, indifference (his least favorite hypothesis), but this? This was absolutely ludicrous. Sherlock could never have anticipated that his best mate would flat out refuse to believe in his existence.

"John, no," Sherlock exclaimed "I never died. I just, had to make you think I did. Then I had to take care of some pressing issues, all of which had to do with taking down part of Moriarty's web!" John just stared incredulously back at Sherlock. Suddenly, he reached over and pinched himself. His face betrayed a quick flit of pain, but was quickly replaced with amazement.

"I'm not dreaming…" he muttered to himself, then looked back up at Sherlock. He cautiously unfurled himself from the huddle mass on the kitchen floor, and cautiously stepped towards his friend. He reached out a shaking hand, and stopped half way between the two, as if afraid to bridge the gap between them, lest he find nothing on the other side. Sherlock confidently met him halfway, gripping John's rough, calloused hand in his smooth, lithe one.

"I have returned John. You asked for one last miracle. Here I am." Sherlock said quietly, a grin breaking out across his face. John stared dumbfounded, as if he could not comprehend that Sherlock was actually there. His cheeks tinged with pink as he met Sherlock's eyes and said "Oh, you heard that then," He paused before continuing "I meant it, every word."

Though Sherlock very rarely felt the need to emote, the vulnerability that John was exuding overrode Sherlock's usually flawless ability to suppress unwanted emotion. He could actually feel the tears prickling at the corners of his eyes as he thought back to John's graveside revelations. What his friend had said had touched Sherlock, but as he couldn't tell John at the time, he had repressed the emotions. Now they came flooding back.

"John, I greatly appreciate what you said. You truly are my best friend, and I sincerely hope that you will be able to forgive me for keeping you in the dark throughout this whole affair. I understand if you are angry, and if you want me to leave, I will respect your wishes. However, John, just know that I was utterly lost without my blogger." Sherlock started bravely, but by the end of his little speech, he looked a bit like a kicked puppy. He meant what he said, but he was fearful that John might make him leave. He started worrying about where he could go and how he would cope knowing that his best friend hated him, and was quickly lost in thought. Consequently, he didn't notice John make a decision.

One moment Sherlock was working his way toward a full blown panic attack. The next, he was on the floor. He blinked rapidly, confused as to how he got there and why the left side of his face suddenly hurt. He looked up to John, who was shaking his hand, looking utterly fulfilled.

"That," John said when he noticed Sherlock looking, "was for leaving and lying to me." Sherlock sheepishly smiled at him. He still wasn't sure whether he would be allowed to stay. Then John stuck his other hand-the one that hadn't just punched Sherlock in the face- out. Sherlock grabbed it and pulled himself erect, directing a questioning gaze to John. As soon as he was up, John pulled him into a rough bear hug. He felt like a consulting detective plush subjected to exposure to an overexcited, small child. He could practically feel his eyes popping out of his skull with the force of the hug.

"And this," John chuckled into Sherlock's ear, "is for coming back."


	3. Present Day

"John! JOHN! JAAAAAWNNNNNN!" The consulting detective yelled as he bolted through the door with a large, heavy envelope in his hands. He seemed greatly distressed, which worried John. He had only seen his friend this emotional a few times, and none of those situations had turned out well. Sherlock ran right up to John and thrust the packet at him. John took it and turned it over in his hands. All the blood left his face and his hands began trembling when he saw the seal that had held the envelope closed. It was achingly familiar, but it couldn't possibly mean what he thought.

John turned the envelope over and let the contents slide out into his hands. John studied them, and then looked up at Sherlock. Both men wore puzzled expressions, but John was the one to say "What do these mean, Sherlock." For once, the detective didn't know. The envelope contained just three items. The first was a post card. It had a picture of a group of islands that were rather strangely shaped. The first was simply a straight line, the second was oval shaped, and the third was similar to the second, but with a massive cove. The islands looked vaguely like-

"IOU" The consulting detective whispered. John just nodded in agreement. It was the fourth Island of the picture that was truly frightening. It was the largest, and mostly circular. But on the island, two buildings and a mountain range seemed to transform the island into a smiley face, not dissimilar to the one on their own flat's wall. The smiley that Sherlock had painted- and shot- on the wall of their flat.

The islands seemed to be connected to each other by bridges, and you could just barely make out pathways on the island. In bright yellow block letters, the top of the postcard said "welcome to your doom!" in all caps. The contrast between the cheery appearance and the actual letters sent a shiver down John's spine. This was from someone with a very twisted sense of humor.

"Turn it over." Sherlock urged. With shaking hands, John did just that. On the back, loopy handwriting covered half of the post card, while the other half had information about the picture. John elected to read the information first, hoping to save the frightening part for last. The paragraph read:

**Welcome to the lovely Reichenbach Isles. This group of islands is a little known archipelago located in the Pacific Ocean. The only known inhabitants of the islands were eaten when a division of the Baskerville facility opened on the main island. The scientists produced a deadly new breed known only as the "water roachinkies". This breed is a cross between a cockroach, tardigrade- informally known as a water bear- and a Twinkie. After the hoard ate all humans on the islands, a private collector purchased the island and started cultivating the stock of water roachinkies. If this heard ever managed to get to mainland, the effects would be truly horrifying.**

"Why would anyone be that stupid?" John questioned. Sherlock, once again, shrugged. John then turned his attention to the hand written message. It proved to be just as chilling as John expected.

**Water Roachinkies are red,**

**The ocean is blue.**

**If you want Mrs. Hudson back, **

**Fly here, will the two of you.**

**-JM**

"No…"John whispered, mortified. He couldn't be back, and he couldn't have Mrs. Hudson. Sherlock just nodded at him.

"Just because the postcard is signed 'JM', it's not necessarily him. More likely it is someone independent of his main criminal web who was given instructions to fulfill this plan should I defy death and return. She is gone; I already checked." Sherlock whispered. John looked down at the other two items from the envelope. They were plane tickets. Stapled to one was directions on how they were to get to the islands. John lifted his gaze to Sherlock's.

"We have to go." They said at the same time. Immediately the two whirled away from each other, collecting a few necessities from around the flat. The tickets were for later that night, so they hurried to get ready in time. When they both were ready, each had a small bag packed with food, water, some clothes, medical supplies and a few other things they thought might come in handy. As they rushed out of the flat, John caught a small smile on Sherlock's face. When they were comfortably settled in the cab and Sherlock had given the address listed on the directions, John sucked in a breath and confronted his friend.

"Sherlock, this is really quite dangerous. It isn't a homicide, where the bad has already happened. If anything bad happens in this, it's going to happen in the future, most likely to us. You're intelligent enough to deduce this by yourself. How can you be so excited?" While John knew most of Sherlock's stranger habits, they never failed to amaze him.

"Ah, my dear Watson," Sherlock said with a smirk, "My body is merely transport, and while I wouldn't like for anything to happen to you, or Mrs. Hudson, I am confident that we can handle anything a Moriarty-wannabe can throw at us. Besides, the game, at long last, is on!"

Watson didn't share his partner's sentiment, but he suppressed his insecurities. Sherlock would need him more than ever on this case. That is, if they both wanted out alive.


	4. To the Island

"John! JOHN! JAAAAAWNNNNNN!" The consulting detective yelled as he bolted through the door with a large, heavy envelope in his hands. He seemed greatly distressed, which worried John. He had only seen his friend this emotional a few times, and none of those situations had turned out well. Sherlock ran right up to John and thrust the packet at him. John took it and turned it over in his hands. All the blood left his face and his hands began trembling when he saw the seal that had held the envelope closed. It was achingly familiar, but it couldn't possibly mean what he thought.

John turned the envelope over and let the contents slide out into his hands. John studied them, and then looked up at Sherlock. Both men wore puzzled expressions, but John was the one to say "What do these mean, Sherlock." For once, the detective didn't know. The envelope contained just three items. The first was a post card. It had a picture of a group of islands that were rather strangely shaped. The first was simply a straight line, the second was oval shaped, and the third was similar to the second, but with a massive cove. The islands looked vaguely like-

"IOU" The consulting detective whispered. John just nodded in agreement. It was the fourth Island of the picture that was truly frightening. It was the largest, and mostly circular. But on the island, two buildings and a mountain range seemed to transform the island into a smiley face, not dissimilar to the one on their own flat's wall. The smiley that Sherlock had painted- and shot- on the wall of their flat.

The islands seemed to be connected to each other by bridges, and you could just barely make out pathways on the island. In bright yellow block letters, the top of the postcard said "welcome to your doom!" in all caps. The contrast between the cheery appearance and the actual letters sent a shiver down John's spine. This was from someone with a very twisted sense of humor.

"Turn it over." Sherlock urged. With shaking hands, John did just that. On the back, loopy handwriting covered half of the post card, while the other half had information about the picture. John elected to read the information first, hoping to save the frightening part for last. The paragraph read:

**Welcome to the lovely Reichenbach Isles. This group of islands is a little known archipelago located in the Pacific Ocean. The only known inhabitants of the islands were eaten when a division of the Baskerville facility opened on the main island. The scientists produced a deadly new breed known only as the "water roachinkies". This breed is a cross between a cockroach, tardigrade- informally known as a water bear- and a Twinkie. After the hoard ate all humans on the islands, a private collector purchased the island and started cultivating the stock of water roachinkies. If this heard ever managed to get to mainland, the effects would be truly horrifying.**

"Why would anyone be that stupid?" John questioned. Sherlock, once again, shrugged. John then turned his attention to the hand written message. It proved to be just as chilling as John expected.

**Water Roachinkies are red,**

**The ocean is blue.**

**If you want Mrs. Hudson back, **

**Fly here, will the two of you.**

**-JM**

"No…"John whispered, mortified. He couldn't be back, and he couldn't have Mrs. Hudson. Sherlock just nodded at him.

"Just because the postcard is signed 'JM', it's not necessarily him. More likely it is someone independent of his main criminal web who was given instructions to fulfill this plan should I defy death and return. She is gone; I already checked." Sherlock whispered. John looked down at the other two items from the envelope. They were plane tickets. Stapled to one was directions on how they were to get to the islands. John lifted his gaze to Sherlock's.

"We have to go." They said at the same time. Immediately the two whirled away from each other, collecting a few necessities from around the flat. The tickets were for later that night, so they hurried to get ready in time. When they both were ready, each had a small bag packed with food, water, some clothes, medical supplies and a few other things they thought might come in handy. As they rushed out of the flat, John caught a small smile on Sherlock's face. When they were comfortably settled in the cab and Sherlock had given the address listed on the directions, John sucked in a breath and confronted his friend.

"Sherlock, this is really quite dangerous. It isn't a homicide, where the bad has already happened. If anything bad happens in this, it's going to happen in the future, most likely to us. You're intelligent enough to deduce this by yourself. How can you be so excited?" While John knew most of Sherlock's stranger habits, they never failed to amaze him.

"Ah, my dear Watson," Sherlock said with a smirk, "My body is merely transport, and while I wouldn't like for anything to happen to you, or Mrs. Hudson, I am confident that we can handle anything a Moriarty-wannabe can throw at us. Besides, the game, at long last, is on!"

Watson didn't share his partner's sentiment, but he suppressed his insecurities. Sherlock would need him more than ever on this case. That is, if they both wanted out alive.


	5. Sine Island

Sometime towards the beginning of the 12 hour flight, a miracle happened. The planets aligned, causing a magnificent sight to behold. Sherlock fell asleep. John followed soon after, not able to bear the silence of the plane for too long. When John awoke, Sherlock was just rousing.

"We are starting our descent into the Moran airstation, please fasten your seatbelts and return your tray tables to their upright and lock position. We hope you have enjoyed your flight on Bartholomew airlines, and please remember, Welcome to Your Doom!" said a voice over the loudspeaker. It was a tenor timbered voice with an irish lilt to it. Both men froze immediately as soon as they heard it.

"Moriarty? But how!" John whispered furiously to Sherlock.

"It's a recording, John. Don't be so dull." Sherlock said. John could see the uncertainty on Sherlock's face, but decided to ignore it for now. Both men collected their bags and disembarked to the Isles. The pilot stepped out behind them, handed Sherlock an envelope then hurried back onto the plane. Sherlock tore open the envelope, calmly scanned the contents, then passed them over to John. It was a simple note.

**To find your beloved , you must make your way from island to island until you reach the volcano located on the main isle. Each island is set us as a giant labyrinth with questions at every turn, literally. You see, I did some digging Sherlock, and discovered that you failed one class in all of your time at school. One measly little concept that you couldn't grasp. Trigonometry. For whatever reason, your brilliant mind couldn't retain this information, and now I will turn that weakness against you. In this maze, only math can show you the way to the next isle. Before you can cross to the next, you shall have to answer one last question about the isle. You may skip over the math and find your own way through the labyrinth, but be warned, the longer you take, the less of a chance of survival has. In roughly twenty four hours, the volcano on the main island will explode and take Mrs. Hudson with it. If you get to her in time and solve the final problem, a motorboat will be idling in the river that lies close by for your escape. Good luck with the trigonometry! Not. This, Sherlock, is how I will BURN the heart out of you!**

**-JM **

John glanced over to see a very irritated Sherlock.

"The one subject!" Sherlock shouted in fury, "I need time to go to my Mind Palace, see if I can dredge up anything from that class."

"But Sherlock-" John tried.

"No John, don't disrupt me. I need you to stop thinking and-" Sherlock exclaimed.

"Sherlock, just listen-" John interrupted.

"be quite. It is of utmost importance that we get through this maze quickly and-" Sherlock talked over John.

"SHERLOCK!" John yelled, finally halting the detective's rant, "Trigonometry was my favorite class. I even brought my book on trigonometry and trigonometric concepts with me," John said, the sheepishly admitted, "reading through it helps me calm down, plus I figured we could use it to you know, determine distances or angles or things like that. I used to do that when I would go on hikes, but I haven't gone since I got back because, y'know, my leg." Sherlock just stared for a moment, then whispered "Brilliant."

"What was that?" John asked cockily.

"Now is hardly the time John I suggest-" Sherlock rushed.

"Hmm, what was that Sherlock?" John egged.

"Brilliant, alright?"Sherlock mumbled grumpily, then, without warning, strode off towards the opening of the labyrinth. John chuckled then followed.

The entrance lead to a long hallway that stretched out on either side of them. right in front of them, they saw word carved into the wall. The uppermost carvings were obviously a math problem. Below that, was two separate groups of carvings. At the bottom of each group was an arrow, each pointing in the opposite direction from each other. Just above them was more math.

"Ahhh, just as I suspected," Sherlock said as he walked up to the wall and traced the carvings with his fingers, "The answer to the math problem will indicate which arrow to follow. Simple enough. Lets see what we can do." Sherlock stepped back to observe the carvings.

**What is the ratio for sine and its cofunction?**

←_-sin = opposite over hypotenuse csc = hypotenuse over opposite_

_OR_

_sin = adjacent over hypotenuse csc = hypotenuse over adjacent -_

"Alright, simple enough," John said, secretly enjoying the baffled look on Sherlock's face, " We need to go left!"

"How could you possibly know that!" Sherlock exclaimed, obviously frustrated.

"It's quite simple really. I always remember sine, cosine and tangent with the acronym SOHCAHTOA, and from there, it's easy." John explained, secretly pleased that Sherlock didn't know this. In fact, Sherlock looked quite lost. _Must be how I usually look_ John thought. He decided to take pity on Sherlock and pulled out his pad of paper and a pencil. He drew a right triangle, then christened the sides and angles. Sherlock watch closely, eager to unlock the secrets of this most baffling subject.

"Alright," John said, once he had drawn everything he needed, "SOHCAHTOA. It stands for sine, opposite, hypotenuse, cosine, adjacent, hypotenuse, tangent, opposite, adjacent. The sine of a given angle is equal to the length of the opposite angle divided by the length of the hypotenuse. The cosine is equal to the length of the adjacent side divided by the length of the hypotenuse. Sense a pattern here?" Sherlock slowly nodded.

"So if this side is 3 units long, this side is 4 units long, then the hypotenuse would be-" John said labeling each side as he went.

"5. That is simplistic enough. Pythagorean Theorem, John. A2 + B2=C2, I can do that." Sherlock interjected. John chuckled under his breath, but let the detective have this small victory.

"Right. So what would the sine of this triangle be?" John asked. Sherlock studied the illustration for a moment before answering.

"According to which angle?" He asked.

"Ummm, angle A." John said.

"Then it would be ⅗," Sherlock said confidently " and before you ask, the cosine would be ⅘, and tangent would be ¾." John smiled approvingly.

"Excellent Sherlock! Now for the cofunctions. Sine and cosecant go together, Cosine and secant go together, and tangent and cotangent go together. Cotangent and tangent are easy enough to remember, the hard part is keeping the other four straight. I just remember that there is a co- in each group and ones that should go together don't." John said. Sherlock nodded in understanding.

"I think I understand. We need to move now, and we can solidify this concept later." Sherlock said as he dashed to the left.

"Wait up Sherlock! My legs aren't as long as yours!" John moaned, but rushed after the detective. It wasn't too long before they came to the next problem. Sherlock rushed up to it immediately, eager to impress John with his new knowledge.

**If sin(-x) = -sinx and csc(-x) = -csc(x), are the functions **

_-Even_

_OR_

_Odd →_

Sherlock's shoulders slumped. This was new, and he had no clue. He gave a defeated sigh, then moved out of the way so John could see the problem.

"Oo, this one is tricky," John said, tapping his foot as he looked at the problem, " I think I need to check the book, this is one thing I don't remember well." John pulled out the book and turned to the appropriate section. He read through the section then looked back at the problem.

"Alright, the answer is odd," John said, "Sine, cosecant, tangent, and cotangent are odd, while cosine and secant are even." Sherlock just nodded. The two men raced to the right. They continued this way, John answering the questions and teaching Sherlock about the trigonometric concepts they of the questions they encountered were found in a similar form as the first two, and included questions such as:

**What is sin(A plus or minus B) equal to?**

_sinAcosB plus or minus cosAsinB -_

_OR_

_-cosAcosB plus or minus sinAsinB_

which led the men to the right, and

**What is plus or minus the square root of 1 - cos(x) over 2?**

_-sin(x/2)_

_OR_

_cos(x/2)-_

which led them to the left. As the day wore on, though, they began encountering a new type of problem.

"We must be getting close to the next island." Sherlock said as they strode up to a massive gate. John nodded, knowing that Sherlock 's assumption would be correct. They approached the gate slowly, looking for any sign of trouble. As they inspected the gate, they found a touch screen to the side of door. John firmly placed his hand upon the screen, which instantly lit up. Another questions was showing on the screen.

**1)If Sin^2+Cos^2=1, then what does Sin^2 equal to?**

"John this isn't trigonometry, this is simple algebra!" Sherlock exclaimed. He eagerly ran up to the computer screen and brought up the keyboard to type in _1 -cos^2._

"Good Sherlock, but the point of that question is to know the trigonometric identities." The next question flashed up on the screen.

**2)Enter the equivalent of sinA plus sinB**

Sherlock just stared at this question. "Is this one even in English!?" he half joked. John gave a small chuckle, the racked his brain for the answer. He had a small moment of panic when he couldn't remember the answer.

"Let's break out the book for this one Sherlock, just incase."He said, not wanting Sherlock to see that he didn't know the answer to this problem. Sherlock grabbed the book from John's backpack thrust it into John's hands.

"I wouldn't know where to look." He admitted sheepishly. John flipped to the correct section. After a small noise of victory, John shut the book and thrust it into Sherlock's hands., who in turn put it back into John's pack. John entered the answer into the computer:

_2sin(A+B/2)cos(A-B/2)_

The next question popped up on the screen. John sighed in relief as the next question popped up on the page. This one had been in the chapter he had just looked at.

**3)Enter the equivalent of sinA minus sinB**

John quickly typed in the answer to this question.

_2cos(A+B/2)sin(A-B/2)_

"I wonder how many more questions there will be." John mused as another question flashed on the screen.

**4)Enter the value for sinAsinB**

Another question that came from the same chapter. John was sensing a pattern here. He quickly entered the answer to this question.

_½[cos(A-B) - cos(A+B)]_

Yet another question popped up onto the screen.

**5)Enter the equation for sinAcosB**

John sighed, then typed in the answer for this question.

_½[sin(A+B)/2) + sin(A - B/2)]_

"This is tedious," John commented as a new question flashed on the screen, "Is this how you feel all the time?" Sherlock just glared at John, not happy that the tables had turned, that he was now the idiot. John chuckled at Sherlock's disgruntled countenance, then got to the question.

**6)Enter the value for cosAsinB**

"Once again, simple!" John exclaimed as he typed in the answer.

_½[cos(A+B) - sin(A-B)]_

After John pressed enter, the computer screen went blank. The two men stared at each other, confused. John poked the screen, but nothing happened. Suddenly, a loud creak emanated from the gate, and it swung open.

"Finally! John exclaimed, then dashed through the door with Sherlock close on his heels. The path was obvious and easy to follow. They ran along it for about ten minutes before breaking out of the dense jungle they had been in since the first problem. The duo found themselves on a beach now. They could see an island to their left, to far away for them to swim, but close enough to tell that it was the next island on their way to the volcano. They could see a bridge further down the beach with another huge gate guarding it.

"Almost there Sherlock, almost to the next island." How long has it taken us so far?" John asked.

Sherlock stopped and pulled his pack off of his back. He rooted through the pack before pulling out a watch.

"We started out roughly six hours ago, therefore, we have roughly eighteen hours left." Sherlock said after a glance at the watch. He then strapped the watch to his wrist .

"We've been at this for six hours already?"John asked. Sherlock nodded. "We need to hurry." they said simultaneously. They picked up their pace as they trotted towards the gate.

"I wonder how many questions this one will be." John pondered aloud as they drew nearer. When they reached the gate, they found, to no surprise, that it was set up much like the last one. John pressed his hand to the touch screen, ready to answer six more questions. To his surprise, one ununmbered question appeared on the screen.

**In order to move on enter the Law of Sines and Law of Cosines**

Of course this question would be harder than the other ones. They were attempting to pass from one island to another and this question would undoubtedly be more difficult than ones they had faced so far.

"Sherlock, look these up for me in the trig book. I think I remember them, but I want to get everything right." John said as he started typing in what he thought was the correct answer.

"John, we have a problem." Sherlock said moments later. John whirled around to see Sherlock holding the book open to what should have been chapter seven of the trig book, but instead was only a few jagged page edges and chapter eight.

"What?! How on Earth?" John said. Then it hit him. He had lent Harry his book when she had to take trigonometry in college, just last semester. She was going back to school and trying to turn her life around; John had just wanted to help her out and loan her some books so she wouldn't have to pay for all of them. But why had she done this? Did she get drunk one night while studying, or just get angry at the chapter? It didn't matter now, all that mattered was that John was going to have to remember this now. He hesitantly typed in what he thought was the answer.

a/sinA = b/sinB = c/sinC

a^2 = b^2 + c^2 - 2bccosA

b^2 = a^2 + c^2 - 2 accosB

c^2 = a^2 + b^2 - 2abcosC

Sherlock looked over John's shoulder as he was typing it in. He nodded when John was done.

"Looks good to me." He said. John puffed up a bit at the detective's words, but then remembered that Sherlock knew nothing about trig, and consequently, John could have typed in anything and Sherlock would have responded the same. John just shook his head, the resignedly pressed enter, hoping that his memory was as good as it used to be. All was still for what seemed like an eternity, but in reality could only have been a minute at most. Then the gates swung effortlessly open without any sound. Sherlock and John gazed at each other triumphantly. They had made it through the first island, now there were just two more to go. The two men stepped out onto the bridge and hurried across to the next island, wondering what challenges it might hold in store.


End file.
